Hans Lindblad (Baltimore)
The free boundary problem for a slightly compressible liquid
Abstract:
We prove a new type of energy estimates for the
compressible Euler's equation with free boundary, with a boundary part and
an interior part. These can be thought of as a generalization of the
energies in Christodoulou and Lindblad [CL] to the compressible case
and do not require the fluid to be irrotational. In addition, we show that
our estimates are in fact uniform in the sound speed. As a
consequence, we obtain convergence of solutions of compressible Euler
equations with a free boundary to solutions of the incompressible
equations,
generalizing the result of Ebin [Eb] to when you have a free
boundary. In the incompressible case our energies reduces to those in
[CL] and our proof in particular gives a simplified proof of the
estimates in [CL] with improved error estimates. Since for an
incompressible irrotational liquid with free surface there are small data
global existence results our result leaves open the possibility of long
time existence also for slightly compressible liquids
with a free surface. This is joint work with Chenyun Luo.